A survey of the different types of vector space partitions

TL;DR

This paper provides a comprehensive survey of vector space partitions, exploring their mathematical relations, known classifications, and recent results, including new necessary conditions and bounds for their existence.

Contribution

It compiles and discusses existing results on vector space partitions, introduces new necessary conditions, and summarizes recent classifications for spaces of small dimension.

Findings

Relations between vector space partitions and other mathematical areas

Recent classifications of partitions in spaces V(n,2) for n ≤ 8

New necessary conditions for the existence of vector space partitions

Abstract

A {\it vector space partition} is here a collection $\mathcal P$ of subspaces of a finite vector space $V(n,q)$, of dimension $n$ over a finite field with $q$ elements, with the property that every non zero vector is contained in a unique member of $\mathcal P$. Vector space partitions relates to finite projective planes, design theory and error correcting codes. In the first part of the talk I will discuss some relations between vector space partitions and other branches of mathematics. The other part of the talk contains a survey of known results on the type of a vector space partition, more precisely: the theorem of Beutelspacher and Heden on $\mathrm{T}$-partitions, rather recent results of El-Zanati et al. on the different types that appear in the spaces V(n,2), for $n\leq8$, a result of Heden and Lehmann on vector space partitions and maximal partial spreads including their new necessary condition for the existence of a vector space partition, and furthermore, I will give a theorem of Heden on the length of the tail of a vector space partition. Finally, I will also give a few historical remarks.